$\large{\lim\limits_{n\rightarrow +\infty}n^2\left [ e-\left(1+\frac{1}{n}\right)^n\right ]}$
Si prova che $\forall \ n\geq 1$ vale
$$e-\left (1+\frac{1}{n}\right )^n\geq\frac{e}{2n+2}$$
Pertanto
$$n^2\left [e-\left (1+\frac{1}{n}\right )^n\right ]\geq n^2\frac{e}{2n+2}\stackrel{n\to +\infty}{\longrightarrow} +\infty$$
e
$$n\left [e-\left (1+\frac{1}{n}\right )^n\right ]\geq n\frac{e}{2n+2}\stackrel{n\to +\infty}{\longrightarrow} \frac{e}{2}$$
